Tài liệu Tuyển tập đề thi IMO thế giới 1998-1999 docx

Preface
This book is a continuation of Mathematical Olympiads 1996-1997: Olym-
piad Problems from Around the World, published by the American Math-
ematics Competitions. It contains solutions to the problems from 34 na-
tional and regional contests featured in the earlier book, together with
selected problems (without solutions) from national and regional contests
given during 1998.
This collection is intended as practice for the serious student who
wishes to improve his or her performance on the USAMO. Some of the
problems are comparable to the USAMO in that they came from na-
tional contests. Others are harder, as some countries first have a national
olympiad, and later one or more exams to select a team for the IMO. And
some problems come from regional international contests (“mini-IMOs”).
Different nations have different mathematical cultures, so you will find
some of these problems extremely hard and some rather easy. We have
tried to present a wide variety of problems, especially from those countries
that have often done well at the IMO.
Each contest has its own time limit. We have not furnished this infor-
mation, because we have not always included complete exams. As a rule
of thumb, most contests allow a time limit ranging between one-half to
one full hour per problem.
Thanks to the following students of the 1998 and 1999 Mathematical
Olympiad Summer Programs for their help in preparing and proofreading
solutions: David Arthur, Reid Barton, Gabriel Carroll, Chi-Bong Chan,
Lawrence Detlor, Daniel Katz, George Lee, Po-Shen Loh, Yogesh More,
Oaz Nir, David Speyer, Paul Valiant, Melanie Wood. Without their ef-
forts, this work would not have been possible. Thanks also to Alexander
Soifer for correcting an early draft of the manuscript.
The problems in this publication are copyrighted. Requests for repro-
duction permissions should be directed to:
Dr. Walter Mientka

Secretary, IMO Advisory Board
1740 Vine Street
Lincoln, NE 68588-0658, USA.
Contents
1 1997 National Contests: Solutions 3
1.1 Austria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Bulgaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 Colombia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Czech and Slovak Republics . . . . . . . . . . . . . . . . . . 34
1.7 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.8 Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.9 Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.10 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.11 Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.12 Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.13 Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.14 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.15 Korea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.16 Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1.17 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.18 Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1.19 South Africa . . . . . . . . . . . . . . . . . . . . . . . . . . 105
1.20 Spain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
1.21 Taiwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1.22 Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
1.23 Ukraine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1.24 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . 127
1.25 United States of America . . . . . . . . . . . . . . . . . . . 130

1.26 Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2 1997 Regional Contests: Solutions 141
2.1 Asian Pacific Mathematics Olympiad . . . . . . . . . . . . . 141
2.2 Austrian-Polish Mathematical Competition . . . . . . . . . 145
2.3 Czech-Slovak Match . . . . . . . . . . . . . . . . . . . . . . 149
2.4 Hungary-Israel Mathematics Competition . . . . . . . . . . 153
2.5 Iberoamerican Mathematical Olympiad . . . . . . . . . . . 156
2.6 Nordic Mathematical Contest . . . . . . . . . . . . . . . . . 161
2.7 Rio Plata Mathematical Olympiad . . . . . . . . . . . . . . 163
2.8 St. Petersburg City Mathematical Olympiad (Russia) . . . 166
1
3 1998 National Contests: Problems 180
3.1 Bulgaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.2 Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.3 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.4 Czech and Slovak Republics . . . . . . . . . . . . . . . . . . 185
3.5 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3.6 India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.7 Iran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
3.8 Ireland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
3.9 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.10 Korea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.11 Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.12 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
3.13 Russia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
3.14 Taiwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
3.15 Turkey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
3.16 United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . 209
3.17 United States of America . . . . . . . . . . . . . . . . . . . 211
3.18 Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

4 1998 Regional Contests: Problems 213
4.1 Asian Pacific Mathematics Olympiad . . . . . . . . . . . . . 213
4.2 Austrian-Polish Mathematics Competition . . . . . . . . . . 214
4.3 Balkan Mathematical Olympiad . . . . . . . . . . . . . . . . 216
4.4 Czech-Slovak Match . . . . . . . . . . . . . . . . . . . . . . 217
4.5 Iberoamerican Olympiad . . . . . . . . . . . . . . . . . . . . 218
4.6 Nordic Mathematical Contest . . . . . . . . . . . . . . . . . 219
4.7 St. Petersburg City Mathematical Olympiad (Russia) . . . 220
2
1 1997 National Contests: Solutions
1.1 Austria
1. Solve the system for x, y real:
(x − 1)(y
2
+ 6) = y(x
2
+ 1)
(y −1)(x
2
+ 6) = x(y
2
+ 1).
Solution: We begin by adding the two given equations together.
After simplifying the resulting equation and completing the square,
we arrive at the following equation:
(x − 5/2)
2
+ (y −5/2)
2
= 1/2. (1)

We can also subtract the two equations; subtracting the second given
equation from the first and grouping, we have:
xy(y − x) + 6(x − y) + (x + y)(x − y) = xy(x − y) + (y −x)
(x − y)(−xy + 6 + (x + y) − xy + 1) = 0
(x − y)(x + y −2xy + 7) = 0
Thus, either x − y = 0 or x + y − 2xy + 7 = 0. The only ways to
have x − y = 0 are with x = y = 2 or x = y = 3 (found by solving
equation (1) with the substitution x = y).
Now, all solutions to the original system where x = y will be solutions
to x + y −2xy + 7 = 0. This equation is equivalent to the following
equation (derived by rearranging terms and factoring).
(x − 1/2)(y −1/2) = 15/4. (2)
Let us see if we can solve equations (1) and (2) simultaneously. Let
a = x − 5/2 and b = y −5/2. Then, equation (1) is equivalent to:
a
2
+ b
2
= 1/2 (3)
and equation (2) is equivalent to:
(a+2)(b+2) = 15/4 ⇒ ab+2(a+b) = −1/4 ⇒ 2ab+4(a+b) = −1/2.
(4)
3
Adding equation (4) to equation (3), we find:
(a + b)
2
+ 4(a + b) = 0 ⇒ a + b = 0, −4 (5)
Subtracting equation (4) from equation (3), we find:
(a − b)
2

− 4(a + b) = 1. (6)
But now we see that if a + b = −4, then equation (6) will be false;
thus, a + b = 0. Substituting this into equation (6), we obtain:
(a − b)
2
= 1 ⇒ a − b = ±1 (7)
Since we know that a + b = 0 from equation (5), we now can find
all ordered pairs (a, b) with the help of equation (7). They are
(−1/2, 1/2) and (1/2, −1/2). Therefore, our only solutions (x, y)
are (2, 2), (3, 3), (2, 3), and (3, 2).
2. Consider the sequence of positive integers which satisfies a
n
= a
2
n−1
+
a
2
n−2
+ a
2
n−3
for all n ≥ 3. Prove that if a
k
= 1997 then k ≤ 3.
Solution: We proceed indirectly; assume that for some k > 3,
a
k
= 1997. Then, each of the four numbers a
k−1

, a
k−2
, a
k−3
,
and a
k−4
must exist. Let w = a
k−1
, x = a
k−2
, y = a
k−3
, and
z = a
k−4
. Now, by the given condition, 1997 = w
2
+ x
2
+ y
2
. Thus,
w ≤
√
1997 < 45, and since w is a positive integer, w ≤ 44. But
then x
2
+ y
2

≥ 1997 − 44
2
= 61.
Now, w = x
2
+ y
2
+ z
2
. Since x
2
+ y
2
≥ 61 and z
2
≥ 0, x
2
+ y
2
+
z
2
≥ 61. But w ≤ 44. Therefore, we have a contradiction and our
assumption was incorrect.
If a
k
= 1997, then k ≤ 3.
3. Let k be a positive integer. The sequence a
n
is defined by a

1
= 1, and
a
n
is the n-th positive integer greater than a
n−1
which is congruent
to n modulo k. Find a
n
in closed form.
Solution: We have a
n
= n(2+(n−1)k)/2. If k = 2, then a
n
= n
2
.
First, observe that a
1
≡ 1 (mod k). Thus, for all n, a
n
≡ n
(mod k), and the first positive integer greater than a
n−1
which is
congruent to n modulo k must be a
n−1
+ 1.
4
The n-th positive integer greater than a

n−1
that is congruent to n
modulo k is simply (n − 1)k more than the first positive integer
greater than a
n−1
which satisfies that condition. Therefore, a
n
=
a
n−1
+ 1 + (n − 1)k. Solving this recursion gives the above answer.
4. Given a parallelogram ABCD, inscribe in the angle ∠BAD a circle
that lies entirely inside the parallelogram. Similarly, inscribe a circle
in the angle ∠BCD that lies entirely inside the parallelogram and
such that the two circles are tangent. Find the locus of the tangency
point of the circles, as the two circles vary.
Solution: Let K
1
be the largest circle inscribable in ∠BAD such
that it is completely inside the parallelogram. It intersects the line
AC in two points; let the point farther from A be P
1
. Similarly, let
K
2
be the largest circle inscribable in ∠BCD such that it is com-
pletely inside the parallelogram. It intersects the line AC in two
points; let the point farther from C be P
2
. then the locus is the

intersection of the segments AP
1
and CP
2
.
We begin by proving that the tangency point must lie on line AC.
Let I
1
be the center of the circle inscribed in ∠BAD. Let I
2
be
the center of the circle inscribed in ∠BCD. Let X represent the
tangency point of the circles.
Since circles I
1
and I
2
are inscribed in angles, these centers must
lie on the respective angle bisectors. Then, since AI
1
and CI
2
are
bisectors of opposite angles in a parallelogram, they are parallel;
therefore, since I
1
I
2
is a transversal, ∠AI
1

X = ∠CI
2
X.
Let T
1
be the foot of the perpendicular from I
1
to AB. Similarly,
let T
2
be the foot of the perpendicular from I
2
to CD. Observe that
I
1
T
1
/AI
1
= sin ∠I
1
AB = sin ∠I
2
CD = I
2
T
2
/CI
2
. But I

1
X = I
1
T
1
and I
2
X = I
2
T
2
. Thus, I
1
X/AI
1
= I
2
X/CI
2
.
Therefore, triangles CI
2
X and AI
1
X are similar, and vertical angles
∠I
1
XA and ∠I
2
XC are equal. Since these vertical angles are equal,

the points A, X, and C must be collinear.
The tangency point, X, thus lies on diagonal AC, which was what
we wanted.
Now that we know that X will always lie on AC, we will prove that
any point on our locus can be a tangency point. For any X on our
5
locus, we can let circle I
1
be the smaller circle through X, tangent
to the sides of ∠BAD.
It will definitely fall inside the parallelogram because X is between
A and P
1
. Similarly, we can draw a circle tangent to circle I
1
and
to the sides of ∠BCD; from our proof above, we know that it must
be tangent to circle I
1
at X. Again, it will definitely fall in the
parallelogram because X is between C and P
2
.
Thus, any point on our locus will work for X. To prove that any
other point will not work, observe that any other point would either
not be on line AC or would not allow one of the circles I
1
or I
2
to

be contained inside the parallelogram.
Therefore, our locus is indeed the intersection of segments AP
1
and
CP
2
.
6
1.2 Bulgaria
1. Find all real numbers m such that the equation
(x
2
− 2mx − 4(m
2
+ 1))(x
2
− 4x − 2m(m
2
+ 1)) = 0
has exactly three different roots.
Solution: Answer: m = 3. Proof: By setting the two factors
on the left side equal to 0 we obtain two polynomial equations, at
least one of which must be true for some x in order for x to be a
root of our original equation. These equations can be rewritten as
(x − m)
2
= 5m
2
+ 4 and (x −2)
2

= 2(m
3
+ m + 2). We have three
ways that the original equation can have just three distinct roots:
either the first equation has a double root, the second equation has
a double root, or there is one common root of the two equations.The
first case is out, however, because this would imply 5m
2
+ 4 = 0
which is not possible for real m.
In the second case, we must have 2(m
3
+ m + 2) = 0; m
3
+ m + 2
factors as (m+1)(m
2
−m+2) and the second factor is always positive
for real m. So we would have to have m = −1 for this to occur. Then
the only root of our second equation is x = 2, and our first equation
becomes (x + 1)
2
= 9, i.e. x = 2, −4. But this means our original
equation had only 2 and -4 as roots, contrary to intention.
In our third case let r be the common root, so x − r is a factor of
both x
2
−2mx −4(m
2
+ 1) and x

2
−4x −2m(m
2
+ 1). Subtracting,
we get that x −r is a factor of (2m −4)x−(2m
3
−4m
2
+2m−4), i.e.
(2m−4)r = (2m−4)(m
2
+1). So m = 2 or r = m
2
+1. In the former
case, however, both our second-degree equations become (x −2)
2
=
24 and so again we have only two distinct roots. So we must have
r = m
2
+1 and then substitution into (r −2)
2
= 2(m
3
+m + 2) gives
(m
2
− 1)
2
= 2(m

3
+ m + 2), which can be rewritten and factored
as (m + 1)(m − 3)(m
2
+ 1) = 0. So m = −1 or 3; the first case
has already been shown to be spurious, so we can only have m = 3.
Indeed, our equations become (x − 3)
3
= 49 and (x − 2)
2
= 64 so
x = −6, −4, 10, and indeed we have 3 roots.
2. Let ABC be an equilateral triangle with area 7 and let M, N be
points on sides AB, AC, respectively, such that AN = BM. Denote
7
by O the intersection of BN and CM. Assume that triangle BOC
has area 2.
(a) Prove that MB/AB equals either 1/3 or 2/3.
(b) Find ∠AOB.
Solution:
(a) Let L be on BC with CL = AN , and let the intersections of
CM and AL, AL and BN be P, Q, respectively. A 120-degree
rotation about the center of ABC takes A to B, B to C, C to
A; this same rotation then also takes M to L, L to N, N to
M, and also O to P , P to Q, Q to O. Thus OP Q and M LN
are equilateral triangles concentric with ABC. It follows that
∠BOC = π −∠NOC = 2π/3, so O lies on the reflection of the
circumcircle of ABC through BC. There are most two points
O on this circle and inside of triangle ABC such that the ratio
of the distances to BC from O and from A — i.e. the ratio of

the areas of triangles OBC and ABC — can be 2/7; so once we
show that MB/AB = 1/3 or 2/3 gives such positions of O it will
follow that there are no other such ratios (no two points M can
give the same O, since it is easily seen that as M moves along
AB, O varies monotonically along its locus). If MB/AB =
1/3 then AN/AC = 1/3, and Menelaus’ theorem in triangle
ABN and line CM gives BO/ON = 3/4 so [BOC]/[BNC] =
BO/BN = 3/7. Then [BOC]/[ABC] = (3/7)(CN/CA) =
2/7 as desired. Similarly if MB/AB = 2/3 the theorem gives
us BO/BN = 6, so [BOC]/[BNC] = BO/BN = 6/7 and
[BOC]/[ABC] = (6/7)(CN/AC) = 2/7.
(b) If MB/AB = 1/3 then M ONA is a cyclic quadrilateral since
∠A = π/3 and ∠O = π − (∠P OQ) = 2π/3. Thus ∠AOB =
∠AOM + ∠MOB = ∠ANM + ∠P OQ = ∠ANM + π/3. But
MB/AB = 1/3 and AN/AC = 1/3 easily give that N is the
projection of M onto AC, so ∠ANM = π/2 and ∠AOB =
5π/6.
If MB/AB = 2/3 then MONA is a cyclic quadrilateral as
before, so that ∠AOB = ∠AOM +∠MOB = ∠ANM +∠POQ.
But AMN is again a right triangle, now with right angle at M,
and ∠MAN = π/3 so ∠ANM = π/6, so ∠AOB = π/2.
8
3. Let f(x) = x
2
− 2ax − a
2
− 3/4. Find all values of a such that
|f(x)| ≤ 1 for all x ∈ [0, 1].
Solution: Answer: −1/2 ≤ a ≤
√

2/4. Proof: The graph of f(x)
is a parabola with an absolute minimum (i.e., the leading coefficient
is positive), and its vertex is (a, f(a)). Since f (0) = −a
2
− 3/4, we
obtain that |a| ≤ 1/2 if we want f(0) ≥ −1. Now suppose a ≤ 0;
then our parabola is strictly increasing between x = 0 and x = 1 so
it suffices to check f(1) ≤ 1. But we have 1/2 ≤ a + 1 ≤ 1, 1/4 ≤
(a + 1)
2
≤ 1, 1/4 ≤ 5/4 − (a + 1)
2
≤ 1. Since 5/4 −(a + 1)
2
= f(1),
we have indeed that f meets the conditions for −1/2 ≤ a ≤ 0. For
a > 0, f decreases for 0 ≤ x ≤ a and increases for a ≤ x ≤ 1. So we
must check that the minimum value f(a) is in our range, and that
f(1) is in our range. This latter we get from 1 < (a + 1)
2
≤ 9/4
(since a ≤ 1/2) and so f(x) = −1 ≤ 5/4 − (a + 1)
2
< 1/4. On
the other hand, f(a) = −2a
2
− 3/4, so we must have a ≤
√
2/4 for
f(a) ≥ −1. Conversely, by bounding f (0), f(a), f(1) we have shown

that f meets the conditions for 0 < a ≤
√
2/4.
4. Let I and G be the incenter and centroid, respectively, of a triangle
ABC with sides AB = c, BC = a, CA = b.
(a) Prove that the area of triangle CIG equals |a − b|r/6, where r
is the inradius of ABC.
(b) If a = c + 1 and b = c −1, prove that the lines IG and AB are
parallel, and find the length of the segment IG.
Solution:
(a) Assume WLOG a > b. Let CM be a median and CF be the
bisector of angle C; let S be the area of triangle ABC. Also let
BE be the bisector of angle B; by Menelaus’ theorem on line
BE and triangle ACF we get (CE/EA)(AB/BF )(F I/IC) =
1. Applying the Angle Bisector Theorem twice in triangle
ABC we can rewrite this as (a/c)((a + b)/a)(F I/IC) = 1, or
IC/F I = (a + b)/c, or IC/CF = (a + b)/(a + b + c). Now
also by the Angle Bisector Theorem we have BF = ac/(a + b);
since BM = c/2 and a > b then MF = (a − b)c/2(a + b). So
comparing triangles CMF and ABC, noting that the altitudes
9
to side MF (respectively AB) are equal, we have [CMF]/S =
(a − b)/2(a + b). Similarly using altitudes from M in triangles
CM I and CMF (and using the ratio IC/CF found earlier),
we have [CMI]/S = (a − b)/2(a + b + c); and using altitudes
from I in triangles CGI and CMI gives (since CG/CM = 2/3)
[CGI]/S = (a−b)/3(a + b + c). Finally S = (a+b + c)r/2 leads
to [CGI] = (a − b)r/6.
(b) As noted earlier, IC/CF = (a + b)/(a + b + c) = 2/3 =
CG/CM in the given case. But C, G, M are collinear, as are

C, I, F , giving the desired parallelism (since line M F = line
AB). We found earlier MF = (a − b)c/2(a + b) = 1/2, so
GI = (2/3)(M F ) = 1/3.
5. Let n ≥ 4 be an even integer and A a subset of {1, 2, . . . , n}. Consider
the sums e
1
x
1
+ e
2
x
2
+ e
3
x
3
such that:
• x
1
, x
2
, x
3
∈ A;
• e
1
, e
2
, e
3

∈ {−1, 0, 1};
• at least one of e
1
, e
2
, e
3
is nonzero;
• if x
i
= x
j
, then e
i
e
j
= −1.
The set A is free if all such sums are not divisible by n.
(a) Find a free set of cardinality n/4.
(b) Prove that any set of cardinality n/4+ 1 is not free.
Solution:
(a) We show that the set A = {1, 3, 5, , 2n/4− 1} is free. Any
combination e
1
x
1
+ e
2
x
2

+ e
3
x
3
with zero or two e
i
’s equal
to 0 has an odd value and so is not divisible by n; otherwise,
we have one e
i
equal to 0, so we have either a difference of
two distinct elements of A, which has absolute value less than
2n/4 and cannot be 0, so it is not divisible by n, or a sum
(or negative sum) of two elements, in which case the absolute
value must range between 2 and 4n/4 −2 < n and so again
is not divisible by n.
10
(b) Suppose A is a free set; we will show |A| ≤ n/4. For any k,
k and n −k cannot both be in A since their sum is n; likewise,
n and n/2 cannot be in A. If we change any element k of A to
n −k then we can verify that the set of all combinations

e
i
x
i
taken mod n is invariant, since we can simply flip the sign of
any e
i
associated with the element k in any combination. Hence

we may assume that A is a subset of B = {1, 2, , n/2 − 1}.
Let d be the smallest element of A. We group all the elements
of B greater than d into “packages” of at most 2d elements,
starting with the largest; i.e. we put the numbers from n/2 −
2d to n/2 − 1 into one package, then put the numbers from
n/2 − 4d to n/2 − 2d − 1 into another, and so forth, until we
hit d + 1 and at that point we terminate the packaging process.
All our packages, except possibly the last, have 2d elements; so
let p + 1 be the number of packages and let r be the number
of elements in the last package (assume p ≥ 0, since otherwise
we have no packages and d = n/2 −1 so our desired conclusion
holds because |A| = 1). The number of elements in B is then
2dp+r +d, so n = 4dp+2d+ 2r + 2. Note that no two elements
of A can differ by d, since otherwise A is not free. Also the
only element of A not in a package is d, since it is the smallest
element and all higher elements of B are in packages.
Now do a case analysis on r. If r < d then each complete
package has at most d elements in common with A, since the
elements of any such package can be partitioned into disjoint
pairs each with difference d. Thus |A| ≤ 1 + dp + r and 4|A| ≤
4dp+4r+4 ≤ n (since r+1 ≤ d) so our conclusion holds. If r = d
then each complete package has at most d elements in common
with A, and also the last package (of d elements) has at most
d − 1 elements in common with A for the following reason: its
highest element is 2d, but 2d is not in A since d+d−2d = 0. So
|A| ≤ d(p+1), 4|A| < n and our conclusion holds. If r > d then
we can form r − d pairs in the last package each of difference
d, so each contains at most 1 element of A, and then there
are 2d − r remaining elements in this package. So this package
contains at most d elements, and the total number of elements

in A is at least d(p + 1) + 1, so 4|A| ≤ n and our conclusion
again holds.
11
6. Find the least natural number a for which the equation
cos
2
π(a − x) − 2 cos π(a − x) + cos
3πx
2a
cos

πx
2a
+
π
3

+ 2 = 0
has a real root.
Solution: The smallest such a is 6. The equation holds if a =
6, x = 8. To prove minimality, write the equation as
(cos π(a − x) − 1)
2
+ (cos(3πx/2a) cos(πx/2a + π/3) + 1) = 0;
since both terms on the left side are nonnegative, equality can only
hold if both are 0. From cos π(a − x) − 1 = 0 we get that x is an
integer congruent to a (mod 2). From the second term we see that
each cosine involved must be −1 or 1 for the whole term to be 0; if
cos(πx/2a + π/3) = 1 then πx/2a + π/3 = 2kπ for some integer k,
and multiplying through by 6a/π gives 3x ≡ −2a (mod 12a), while

if the cosine is −1 then πx/2a + π/3 = (2k + 1)π and multiplying by
6a/π gives 3x ≡ 4a (mod 12a). In both cases we have 3x divisible
by 2, so x is divisible by 2 and hence so is a. Also our two cases give
−2a and 4a, respectively, are divisible by 3, so a is divisible by 3.
We conclude that 6|a and so our solution is minimal.
7. Let ABCD be a trapezoid (AB||CD) and choose F on the segment
AB such that DF = CF . Let E be the intersection of AC and BD,
and let O
1
, O
2
be the circumcenters of ADF, BCF . Prove that the
lines EF and O
1
O
2
are perpendicular.
Solution: Project each of points A, B, F orthogonally onto CD to
obtain A

, B

, F

; then F

is the midpoint of CD. Also let the cir-
cumcircles of AF D, BF C intersect line CD again at M, N respec-
tively; then AFMD, BF NC are isosceles trapezoids and F


M =
DA

, NF

= B

C. Let x = DA

, y = A

F

= AF , z = F

B

= F B,
w = B

C, using signed distances throughout (x < 0 if D is be-
tween A

and F

, etc.), so we have x+y = z + w; call this value S, so
DC = 2S. Also let line F E meet DC at G; since a homothety about
E with (negative) ratio CD/AB takes triangle ABE into CDE it
also takes F into G, so DG/GC = F B/AF = F


B

/A

B

= z/y
and we easily get DG = 2zS/(y + z), GC = 2yS/(y + z). Now
12
NF

= w, DF

= S implies DN = z and so DN/DG = (y + z)/2S.
Similarly F

M = x, F

C = S so MC = y and M C/GC = (y +z)/2S
also. So DN/DG = MC/GC, NG/DG = GM/GC and NG ·GC =
DG · GM. Since NC and DM are the respective chords of the
circumcircles of BF C and ADC that contain point G we conclude
that G has equal powers with respect to these two circles, i.e. it is
on the radical axis. F is also on the axis since it is an intersection
point of the circles, so the line F GE is the radical axis, which is
perpendicular to the line O
1
O
2
connecting the centers of the circles.

8. Find all natural numbers n for which a convex n-gon can be di-
vided into triangles by diagonals with disjoint interiors, such that
each vertex of the n-gon is the endpoint of an even number of the
diagonals.
Solution: We claim that 3|n is a necessary and sufficient condition.
To prove sufficiency, we use induction of step 3. Certainly for n = 3
we have the trivial dissection (no diagonals drawn). If n > 3 and 3|n
then let A
1
, A
2
, . . . , A
n
be the vertices of an n-gon in counterclock-
wise order; then draw the diagonals A
1
A
n−3
, A
n−3
A
n−1
, A
n−1
A
1
;
these three diagonals divide our polygon into three triangles and an
(n − 3)-gon A
1

A
2
. . . A
n−3
. By the inductive hypothesis the latter
can be dissected into triangles with evenly many diagonals at each
vertex, so we obtain the desired dissection of our n-gon, since each
vertex from A
2
through A
n−4
has the same number of diagonals in
the n-gon as in the (n −3)-gon (an even number), A
1
and A
n−3
each
have two diagonals more than in the (n − 3)-gon, while A
n−1
has 2
diagonals and A
n
and A
n−2
have 0 each.
To show necessity, suppose we have such a decomposition of a poly-
gon with vertices A
1
, A
2

, . . . , A
n
in counterclockwise order, and for
convenience assume labels are mod n. Call a diagonal A
i
A
j
in our
dissection a “right diagonal” from A
i
if no point A
i+2
, A
i+3
, . . . , A
j−1
is joined to A
i
(we can omit A
i+1
from our list since it is joined
by an edge). Clearly every point from which at least one diagonal
emanates has a unique right diagonal. Also we have an important
lemma: if A
i
A
j
is a right diagonal from A
i
, then within the polygon

A
i
A
i+1
. . . A
j
, each vertex belongs to an even number of diagonals.
Proof: Each vertex from any of the points A
i+1
, . . . , A
j−1
belongs
to an even number of diagonals of the n-gon, but since the diagonals
13
of the n-gon are nonintersecting these diagonals must lie within our
smaller polygon, so we have an even number of such diagonals for
each of these points. By hypothesis, A
i
is not connected via a diag-
onal to any other point of this polygon, so we have 0 diagonals from
A
i
, an even number. Finally evenly many diagonals inside this poly-
gon stem from A
j
, since otherwise we would have an odd number of
total endpoints of all diagonals.
Now we can show 3|n by strong induction on n. If n = 1 or 2, then
there is clearly no decomposition, while if n = 3 we have 3|n. For
n > 3 choose a vertex A

i
1
with some diagonal emanating from it,
and let A
i
1
A
i
2
be the right diagonal from A
i
1
. By the lemma there
are evenly many diagonals from A
i
2
with their other endpoints in
{A
i
1
+1
, A
i
1
+2
, . . . , A
i
2
−1
}, and one diagonal A

i
1
A
i
2
, so there must
be at least one other diagonal from A
i
2
(since the total number of di-
agonals there is even). This implies A
i
1
A
i
2
is not the right diagonal
from A
i
2
, so choose the right diagonal A
i
2
A
i
3
. Along the same lines
we can choose the right diagonal A
i
3

A
i
4
from A
i
3
, with A
i
2
and A
i
4
distinct, then continue with A
i
4
A
i
5
as the right diagonal from A
i
4
,
etc. Since the diagonals of the n-gon are nonintersecting this pro-
cess must terminate with some A
i
k+1
= A
i
1
. Now examine each of

the polygons A
i
x
A
i
x
+1
A
i
x
+2
. . . A
i
x+1
, x = 1, 2, . . . , k (indices x are
taken mod k). By the lemma each of these polygons is divided into
triangles by nonintersecting diagonals with evenly many diagonals
at each vertex, so by the inductive hypothesis the number of vertices
of each such polygon is divisible by 3. Also consider the polygon
A
i
1
A
i
2
. . . A
i
k
. We claim that in this polygon, each vertex belongs
to an even number of diagonals. Indeed, from A

i
x
we have an even
number of diagonals to points in A
i
x−1
+1
, A
i
x−1
+2
, . . . , A
i
x
−1
, plus
the two diagonals A
i
x−1
A
i
x
and A
i
x
A
i
x+1
. This leaves an even num-
ber of diagonals from A

i
x
to other points; since A
i
x
was chosen as
the endpoint of a right diagonal we have no diagonals lead to points
in A
i
x
+1
, . . . , A
i
x+1
−1
, so it follows from the nonintersecting crite-
rion that all remaining diagonals must lead to points A
i
y
for some y.
Thus we have an even number of diagonals from A
i
x
to points A
i
y
for some fixed x; it follows from the induction hypothesis that 3|k.
So, if we count each vertex of each polygon A
i
x

A
i
x
+1
A
i
x
+2
. . . A
i
x+1
once and then subtract the vertices of A
i
1
A
i
2
. . . A
i
k
, each vertex of
our n-gon is counted exactly once, but from the above we have been
adding and subtracting multiples of 3. Thus we have 3|n.
14
9. For any real number b, let f(b) denote the maximum of the function




sin x +

2
3 + sin x
+ b




over all x ∈ R. Find the minimum of f(b) over all b ∈ R.
Solution: The minimum value is 3/4. Let y = 3 + sin x; note
y ∈ [2, 4] and assumes all values therein. Also let g(y) = y + 2/y;
this function is increasing on [2, 4], so g(2) ≤ g(y) ≤ g(4). Thus
3 ≤ g(y) ≤ 9/2, and both extreme values are attained. It now fol-
lows that the minimum of f(b) = max(|g(y) + b −3|) is 3/4, which
is attained by b = −3/4; for if b > −3/4 then choose x = π/2 so
y = 4 and then g(y) + b − 3 > 3/4, while if b < −3/4 then choose
x = −π/2 so y = 2 and g(y) + b −3 = −3/4; on the other hand, our
range for g(y) guarantees −3/4 ≤ g(y) + b − 3 ≤ 3/4 for b = −3/4.
10. Let ABCD be a convex quadrilateral such that ∠DAB = ∠ABC =
∠BCD. Let H and O denote the orthocenter and circumcenter of
the triangle ABC. Prove that H, O, D are collinear.
Solution: Let M be the midpoint of B and N the midpoint of
BC. Let E = AB ∩CD and F = BC ∩ AD. Then EBC and F AB
are isosceles triangles, so EN ∩ FM = 0. Thus applying Pappus’s
theorem to hexagon MCEN AF , we find that G, O, D are collinear,
so D lies on the Euler line of ABC and H, O, D are collinear.
11. For any natural number n ≥ 3, let m(n) denote the maximum num-
ber of points lying within or on the boundary of a regular n-gon of
side length 1 such that the distance between any two of the points
is greater than 1. Find all n such that m(n) = n − 1.
Solution: The desired n are 4, 5, 6. We can easily show that

m(3) = 1, e.g. dissect an equilateral triangle ABC into 4 congruent
triangles and then for two points P, Q there is some corner triangle
inside which neither lies; if we assume this corner is at A then the
circle with diameter BC contains the other three small triangles and
so contains P and Q; BC = 1 so P Q ≤ 1. This method will be
useful later; call it a lemma.
15
On the other hand, m(n) ≥ n − 1 for n ≥ 4 as the following process
indicates. Let the vertices of our n-gon be A
1
, A
2
, . . . , A
n
. Take
P
1
= A
1
. Take P
2
on the segment A
2
A
3
at an extremely small
distance d
2
from A
2

; then P
2
P
1
> 1, as can be shown rigorously, e.g.
using the Law of Cosines in triangle P
1
A
2
P
2
and the fact that the
cosine of the angle at A
2
is nonnegative (since n ≥ 4). Moreover P
2
is on a side of the n-gon other than A
3
A
4
, and it is easy to see that
as long as n ≥ 4, the circle of radius 1 centered at A
4
intersects no
side of the n-gon not terminating at A
4
, so P
2
A
4

> 1 while clearly
P
2
A
3
< 1. So by continuity there is a point P
3
on the side A
3
A
4
with
P
2
P
3
= 1. Now slide P
3
by a small distance d
3
on A
3
A
4
towards A
4
;
another trigonometric argument can easily show that then P
2
P

3
> 1.
Continuing in this manner, obtain P
4
on A
4
A
5
with P
3
P
4
= 1 and
slide P
4
by distance d
4
so that now P
3
P
4
> 1, etc. Continue doing
this until all points P
i
have been defined; distances P
i
P
i+1
are now
greater than by construction, P

n−1
P
1
> 1 because P
1
= A
1
while
P
n−1
is in the interior of the side A
n−1
A
n
; and all other P
i
P
j
are
greater than 1 because it is easy to see that the distance between any
two points of nonadjacent sides of the n-gon is at least 1 with equality
possible only when (among other conditions) P
i
, P
j
are endpoints of
their respective sides, and in our construction this never occurs for
distinct i, j. So our construction succeeds. Moreover, as all the
distances d
i

tend to 0 each P
i
tends toward A
i
, so it follows that
the maximum of the distances A
i
P
i
can be made as small as desired
by choosing d
i
sufficiently small. On the other hand, when n > 6
the center O of the n-gon is at a distance greater than 1 from each
vertex, so if the P
i
are sufficiently close to the A
i
then we will also
have OP
i
> 1 for each i. Thus we can add the point O to our set,
showing that m(n) ≥ n for n > 6.
It now remains to show that we cannot have more than n −1 points
at mutual distances greater than 1 for n = 4, 5, 6. As before let the
vertices of the polygon be A
1
, etc. and the center O; suppose we have
n points P
1

, . . . , P
n
with P
i
P
j
> 1 for i not equal to j. Since n ≤ 6
it follows that the circumradius of the polygon is not greater than
1, so certainly no P
i
can be equal to O. Let the ray from O through
P
i
intersect the polygon at Q
i
and assume WLOG our numbering is
such that Q
1
, Q
2
, . . . , Q
n
occur in that order around the polygon, in
the same orientation as the vertices were numbered. Let Q
1
be on the
side A
k
A
k+1

. A rotation by angle 2π/n brings A
k
into A
k+1
; let it
16
also bring Q
1
into Q

1
, so triangles Q
1
Q

1
O and A
k
A
k+1
O are similar.
We claim P
2
cannot lie inside or on the boundary of quadrilateral
OQ
1
A
k+1
Q


1
. To see this, note that P
1
Q
1
A
k+1
and P
1
A
k+1
Q

1
are
triangles with an acute angle at P
1
, so the maximum distance from
P
1
to any point on or inside either of these triangles is attained
when that point is some vertex; however P
1
Q
1
≤ OQ
1
≤ 1, and
P
1

A
k+1
≤ O
1
A
k+1
≤ 1 (e.g. by a trigonometric argument similar
to that mentioned earlier), and as for P
1
Q

1
, it is subsumed in the
following case: we can show that P
1
P ≤ 1 for any P on or inside
OQ
1
Q

1
, because n ≤ 6 implies that ∠Q
1
OQ

1
= 2π/n ≥ π/3, and so
we can erect an equilateral triangle on Q
1
Q


1
which contains O, and
the side of this triangle is less than A
k
A
k+1
= 1 (by similar triangles
OA
k
A
k+1
and OQ
1
Q

1
) so we can apply the lemma now to show that
two points inside this triangle are at a distance at most 1. The result
of all this is that P
2
is not inside the quadrilateral OQ
1
A
k+1
Q

1
, so
that ∠P

1
OP
2
= ∠Q
1
OP
2
> 2π/n. On the other hand, the label P
1
is not germane to this argument; we can show in the same way that
∠P
i
OP
i+1
> 2π/n for any i (where P
n+1
= P
1
). But then adding
these n inequalities gives 2π > 2π, a contradiction, so our points P
i
cannot all exist. Thus m(n) ≤ n − 1 for n = 4, 5, 6, completing the
proof.
12. Find all natural numbers a, b, c such that the roots of the equations
x
2
− 2ax + b = 0
x
2
− 2bx + c = 0

x
2
− 2cx + a = 0
are natural numbers.
Solution: We have that a
2
− b, b
2
− c, c
2
− a are perfect squares.
Since a
2
−b ≤ (a − 1)
2
, we have b ≥ 2a −1; likewise c ≥ 2b − 1, a ≥
2c − 1. Putting these together gives a ≥ 8a − 7, or a ≤ 1. Thus
(a, b, c) = (1, 1, 1) is the only solution.
13. Given a cyclic convex quadrilateral ABCD, let F be the intersection
of AC and BD, and E the intersection of AD and BC. Let M, N
be the midpoints of AB, CD. Prove that
MN
EF
=
1
2





AB
CD
−
CD
AB




.
17
Solution: Since ABCD is a cyclic quadrilateral, AB and CD
are antiparallel with respect to the point E, so a reflection through
the bisector of ∠AEB followed by a homothety about E with ratio
AB/CD takes C, D into A, B respectively. Let G be the image of
F under this transformation. Similarly, reflection through the bisec-
tor of ∠AEB followed by homothety about E with ratio CD/AB
takes A, B into C, D; let H be the image of F under this trans-
formation. G, H both lie on the reflection of line EF across the
bisector of ∠AEB, so GH = |EG −EH| = EF|AB/CD −CD/AB|.
On the other hand, the fact that ABCD is cyclic implies (e.g. by
power of a point) that triangles ABF and DCF are similar with
ratio AB/CD. But by virtue of the way the points A, B, G were
shown to be obtainable from C, D, F , we have that BAG is also
similar to DCF with ratio AB/CD, so ABF and BAG are con-
gruent. Hence AG = BF, AF = BG and AGBF is a parallelo-
gram. So the midpoints of the diagonals of AGBF coincide, i.e.
M is the midpoint of GF. Analogously (using the parallelogram
CHDF ) we can show that N is the midpoint of HF . But then
MN is the image of GH under a homothety about F with ratio 1/2,

so MN = GH/2 = (EF/2)|AB/CD − CD/AB| which is what we
wanted to prove.
14. Prove that the equation
x
2
+ y
2
+ z
2
+ 3(x + y + z) + 5 = 0
has no solutions in rational numbers.
Solution: Let u = 2x + 3, v = 2y + 3, w = 2z + 3. Then the
given equation is equivalent to
u
2
+ v
2
+ w
2
= 7.
It is equivalent to ask that the equation
x
2
+ y
2
+ z
2
= 7w
2
has no nonzero solutions in integers; assume on the contrary that

(x, y, z, w) is a nonzero solution with |w| + |x| + |y| + |z| minimal.
18
Modulo 4, we have x
2
+ y
2
+ z
2
≡ 7w
2
, but every perfect square is
congruent to 0 or 1 modulo 4. Thus we must have x, y, z, w even,
and (x/2, y/2, z/2, w/2) is a smaller solution, contradiction.
15. Find all continuous functions f : R → R such that for all x ∈ R,
f(x) = f

x
2
+
1
4

.
Solution: Put g(x) = x
2
+1/4. Note that if −1/2 ≤ x ≤ 1/2, then
x ≤ g(x) ≤ 1/2. Thus if −1/2 ≤ x
0
≤ 1/2 and x
n+1

= g(x
n
) for
n ≥ 0, the sequence x
n
tends to a limit L > 0 with g(L) = L; the
only such L is L = 1/2. By continuity, the constant sequence f(x
n
)
tends to f(1/2). In short, f is constant over [−1/2, 1/2].
Similarly, if x ≥ 1/2, then 1/2 ≤ g(x) ≤ x, so analogously f is
constant on this range. Moreover, the functional equation implies
f(x) = f(−x). We conclude f must be constant.
16. Two unit squares K
1
, K
2
with centers M, N are situated in the plane
so that MN = 4. Two sides of K
1
are parallel to the line MN, and
one of the diagonals of K
2
lies on MN. Find the locus of the mid-
point of XY as X, Y vary over the interior of K
1
, K
2
, respectively.
Solution: Introduce complex numbers with M = −2, N = 2.

Then the locus is the set of points of the form −(w + xi) + (y + zi),
where |w|, |x| < 1/2 and |x + y|, |x − y| <
√
2/2. The result is an
octagon with vertices (1 +
√
2)/2 + i/2, 1/2 + (1 +
√
2)i/2, and so on.
17. Find the number of nonempty subsets of {1, 2, . . . , n} which do not
contain two consecutive numbers.
Solution: If F
n
is this number, then F
n
= F
n−1
+ F
n−2
: such
a subset either contains n, in which case its remainder is a subset of
{1, . . . , n−2}, or it is a subset of {1, . . . , n−1}. From F
1
= 1, F
2
= 2,
we see that F
n
is the n-th Fibonacci number.
18. For any natural number n ≥ 2, consider the polynomial

P
n
(x) =

n
2

+

n
5

x +

n
8

x
2
+ ··· +

n
3k + 2

x
k
,
19
where k = 
n−2

3
.
(a) Prove that P
n+3
(x) = 3P
n+2
(x) − 3P
n+1
(x) + (x + 1)P
n
(x).
(b) Find all integers a such that 3
(n−1)/2
divides P
n
(a
3
) for all
n ≥ 3.
Solution:
(a) This is equivalent to the identity (for 0 ≤ m ≤ (n + 1)/3)

n + 3
3m + 2

= 3

n + 2
3m + 2


−3

n + 1
3m + 2

+

n
3m + 2

+

n
3m − 1

,
which follows from repeated use of the identity

a+1
b

=

a
b

+

a
b−1


.
(b) If a has the required property, then P
5
(a
3
) = 10+a
3
is divisible
by 9, so a ≡ −1 (mod 3). Conversely, if a ≡ −1 (mod 3),
then a
3
+ 1 ≡ 0 (mod 9). Since P
2
(a
3
) = 1, P
3
(a
3
) = 3,
P
4
(a
3
) = 6, it follows from (a) that 3
(n−1)/2
divides P
n
(a

3
)
for all n ≥ 3.
19. Let M be the centroid of triangle ABC.
(a) Prove that if the line AB is tangent to the circumcircle of the
triangle AMC, then
sin ∠CAM + sin ∠CBM ≤
2
√
3
.
(b) Prove the same inequality for an arbitrary triangle ABC.
Solution:
(a) Let G be the midpoint of AB, a, b, c the lengths of sides BC,
CA, AB, and m
a
, m
b
, m
c
the lengths of the medians from A, B, C,
respectively. We have

c
2

2
= GA
2
= GM ·GC =

1
3
m
2
c
=
1
12
(2a
2
+ 2b
2
− c
2
),
20
whence a
2
+ b
2
= 2c
2
and m
a
=
√
3b/2, m
b
=
√

3a/2. Thus
sin ∠CAM + sin ∠CBM = K
1
bm
a
+ K
1
am
b
=
(a
2
+ b
2
) sin C
√
3ab
,
where K is the area of the triangle. By the law of cosines,
a
2
+ b
2
= 4ab cos C, so the right side is 2 sin 2C/
√
3 ≤ 2/
√
3.
(b) There are two circles through C and M touching AB; let A
1

, B
1
be the points of tangency, with A
1
closer to A. Since G is the
midpoint of A
1
B
1
and CM/MG = 2, M is also the centroid of
triangle A
1
B
1
C. Moreover, ∠CAM ≤ ∠CA
1
M and ∠CBM ≤
∠CB
1
M. If the angles ∠CA
1
M and ∠CB
1
M are acute, we are
thus reduced to (a).
It now suffices to suppose ∠CA
1
M > 90
◦
, ∠CB

1
M ≤ 90
◦
.
Then CM
2
> CA
2
1
+ A
1
M
2
, that is,
1
9
(2b
2
1
+ 2a
2
1
− c
2
1
) > b
2
1
+
1

9
(2b
2
1
+ 2c
2
1
− a
2
1
),
where a
1
, b
1
, c
1
are the side lengths of A
1
B
1
C. From (a), we
have a
2
1
+ b
2
1
= c
2

1
and the above inequality is equivalent to
a
2
1
> 7b
2
1
. As in (a), we obtain
sin ∠CB
1
M =
b
1
sin ∠B
1
CA
1
a
1
√
3
=
b
1
a
1
√
3


1 −

a
2
1
+ b
2
1
4a
1
b
1

2
.
Setting b
2
1
/a
2
1
= x, we get
sin ∠CB
1
M =
1
4
√
3


14x − x
2
− 1 <
1
4
√
3

2 −
1
49
− 1 =
1
7
,
since x < 1/7. Therefore
sin ∠CAM + sin ∠CBM < 1 + sin ∠CB
1
M < 1 +
1
7
<
2
√
3
.
20. Let m, n be natural numbers and m + i = a
i
b
2

i
for i = 1, 2, . . . , n,
where a
i
and b
i
are natural numbers and a
i
is squarefree. Find all
values of n for which there exists m such that a
1
+a
2
+···+a
n
= 12.
21
Solution: Clearly n ≤ 12. That means at most three of the
m + i are perfect squares, and for the others, a
i
≥ 2, so actually
n ≤ 7.
We claim a
i
= a
j
for i = j. Otherwise, we’d have m + i = ab
2
i
and

m + j = ab
2
j
, so 6 ≥ n − 1 ≥ (m + j) − (m + i) = a(b
2
j
− b
2
i
). This
leaves the possibilities (b
i
, b
j
, a) = (1, 2, 2) or (2, 3, 1), but both of
those force a
1
+ ··· + a
n
> 12.
Thus the a’s are a subset of {1, 2, 3, 5, 6, 7, 10, 11}. Thus n ≤ 4, with
equality only if {a
1
, a
2
, a
3
, a
4
} = {1, 2, 3, 6}. But in that case,

(6b
1
b
2
b
3
b
4
)
2
= (m + 1)(m + 2)(m + 3)(m + 4) = (m
2
+ 5m + 5)
2
−1,
which is impossible. Hence n = 2 or n = 3. One checks that the
only solutions are then
(m, n) = (98, 2), (3, 3).
21. Let a, b, c be positive numbers such that abc = 1. Prove that
1
1 + a + b
+
1
1 + b + c
+
1
1 + c + a
≤
1
2 + a

+
1
2 + b
+
1
2 + c
.
Solution: Brute force! Put x = a + b + c and y = ab + bc + ca.
Then the given inequality can be rewritten
3 + 4x + y + x
2
2x + y + x
2
+ xy
≤
12 + 4x + y
9 + 4x + 2y
,
or
3x
2
y + xy
2
+ 6xy −5x
2
− y
2
− 24x − 3y −27 ≥ 0,
or
(3x

2
y −5x
2
− 12x) + (xy
2
− y
2
− 3x − 3y) + (6xy −9x − 27) ≥ 0,
which is true because x, y ≥ 3.
22. Let ABC be a triangle and M, N the feet of the angle bisectors of
B, C, respectively. Let D be the intersection of the ray MN with
the circumcircle of ABC. Prove that
1
BD
=
1
AD
+
1
CD
.
22
Solution: Let A
1
, B
1
, C
1
be the orthogonal projections of D onto
BC, CA, AB, respectively. Then

DB
1
= DA sin ∠DAB
1
= DA sin ∠DAC =
DA · DC
2R
,
where R is the circumradius of ABC. Likewise DA
1
= DB ·DC/2R
and DC
1
= DA ·DB/2R. Thus it suffices to prove DB
1
= DA
1
+
DC
1
.
Let m be the distance from M to AB or BC, and n the distance
from N to AC or BC. Also put x = DM/MN (x > 1). Then
DB
1
n
= x,
DC
1
m

= x − 1,
DA
1
− m
n − m
= x.
Hence DB
1
= nx, DC
1
= m(x − 1), DA
1
= nx − m(x − 1) =
DB
1
− DC
1
, as desired.
23. Let X be a set of cardinality n + 1 (n ≥ 2). The ordered n-
tuples (a
1
, a
2
, . . . , a
n
) and (b
1
, b
2
, . . . , b

n
) of distinct elements of X
are called separated if there exist indices i = j such that a
i
= b
j
.
Find the maximal number of n-tuples such that any two of them are
separated.
Solution: If A
n+1
is the maximum number of pairwise separated
n-tuples, we have A
n+1
≤ (n+ 1)A
n
for n ≥ 4, since among pairwise
separated n-tuples, those tuples with a fixed first element are also
pairwise separated. Thus A
n
≤ n!/2. To see that this is optimal,
take all n-tuples (a
1
, . . . , a
n
) such that adding the missing member
at the end gives an even permutation of {1, . . . , n − 1}.
23
1.3 Canada
1. How many pairs (x, y) of positive integers with x ≤ y satisfy gcd(x, y) =

5! and lcm(x, y) = 50!?
Solution: First, note that there are 15 primes from 1 to 50:
(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47).
To make this easier, let’s define f(a, b) to be the greatest power of b
dividing a. (Note g(50!, b) > g(5!, b) for all b < 50.)
Therefore, for each prime p, we have either f(x, p) = f(5!, p) and
f(y, p) = f(50!, p) OR f(y, p) = f(5!, p) and f(x, p) = f(50!, p).
Since we have 15 primes, this gives 2
15
pairs, and clearly x = y in
any such pair (since the gcd and lcm are different), so there are 2
14
pairs with x ≤ y.
2. Given a finite number of closed intervals of length 1, whose union
is the closed interval [0, 50], prove that there exists a subset of the
intervals, any two of whose members are disjoint, whose union has
total length at least 25. (Two intervals with a common endpoint are
not disjoint.)
Solution: Consider
I1 = [1 + e, 2 + e], I2 = [3 + 2e, 4 + 2e], . . . I24 = [47 + 24e, 48 + 24e]
where e is small enough that 48+24e < 50. To have the union of the
intervals include 2k + ke, we must have an interval whose smallest
element is in Ik. However, the difference between an element in Ik
and Ik + 1 is always greater than 1, so these do not overlap.
Taking these intervals and [0, 1] (which must exist for the union to be
[0, 50]) we have 25 disjoint intervals, whose total length is, of course,
25.
3. Prove that
1
1999

<
1
2
·
3
4
· ··· ·
1997
1998
<
1
44
.
24